Age-specific copula-AR-GARCH mortality models. (English) Zbl 1314.91143

Summary: In this paper, we propose AR-GARCH (autoregression-generalized autoregressive conditional heteroskedasticity) models to fit and forecast mortality rates for a given age by two alternative approaches. Specifically, one approach is to fit a time series of mortality rates for some age to an AR(\(n\))-GARCH(1,1) model, and project the mortality rate for that age in the next \(n\)th year; the other is to fit an AR(1)-GARCH(1,1) model, and project the mortality rates recursively for the age in the next consecutive years. Further, we employ the copula method to capture the inter-age mortality dependence. Adopting mortality data of Japan, the UK, and the USA, we demonstrate that it is indispensable to consider the conditional heteroskedasticity in our mortality models which provide better performances in out-of-sample projection and prediction intervals with a higher degree of coverage than the Lee-Carter model. Finally, we numerically illustrate with mortality data of Japan that VaR (Value at Risk) values for longevity risk, regarded as additional reserves for annuity or pension providers, will be overestimated if the conditional heteroskedasticity or/and the inter-age mortality dependence structure are ignored.


91B30 Risk theory, insurance (MSC2010)
62P20 Applications of statistics to economics
91B84 Economic time series analysis
91D20 Mathematical geography and demography
Full Text: DOI


[1] Alexandra, D.; Paul, E., Modeling exchange rate dependence dynamics at different time horizons, J. Int. Money Finance, 29, 1687-1705, (2010)
[2] Cairns, A. J.G., Robust hedging of longevity risk, J. Risk Insurance, 80, 621-648, (2013)
[3] Cairns, A. J.G.; Blake, D.; Dowd, K., A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration, J. Risk Insurance, 73, 687-718, (2006)
[4] Cairns, A. J.G.; Dowd, K.; Blake, D.; Coughlan, G. D., Longevity hedge effectiveness: a decomposition, Quant. Finance, 14, 217-235, (2014) · Zbl 1294.91072
[5] Cox, S. H.; Lin, Y.; Tian, R.; Zuluaga, L. F., Mortality portfolio risk management, J. Risk Insurance, 80, 853-890, (2013)
[6] D’Amato, V.; Haberman, S.; Piscopo, G.; Russolillo, M., Modelling dependent data for longevity projections, Insurance Math. Econom., 51, 694-701, (2012) · Zbl 1285.91054
[7] Engle, R. F., Dynamic conditional correlation—a simple class of multivariate GARCH, J. Bus. Econom. Statist., 20, 339-350, (2002)
[8] Garcia, R.; Tsafack, G., Dependence structure and extreme comovements in international equity and bond markets, J. Bank. Finance, 35, 8, 1954-1970, (2011)
[9] Guégan, D.; Zang, J., Change analysis of a dynamic copula for measuring dependence in multivariate financial data, Quant. Finance, 10, 421-430, (2010) · Zbl 1203.91311
[10] Haberman, S.; Renshaw, A., On age-period-cohort parametric mortality rate projections, Insurance Math. Econom., 45, 255-270, (2009) · Zbl 1231.91195
[11] Haberman, S.; Renshaw, A., A comparative study of parametric mortality projection models, Insurance Math. Econom., 48, 35-55, (2011)
[12] Hári, N.; Waegenaere, A.; Melenberg, B.; Nijman, T. E., Longevity risk in portfolios of pension annuities, Insurance Math. Econom., 42, 505-519, (2008) · Zbl 1152.91586
[13] Joe, H.; Xu, J. J., The estimation method of inference functions for margins for multivariate models, technical report, 166, (1996), Department of Statistics, University of British Columbia
[14] Kenourgios, D.; Samitas, A.; Paltalidis, N., Financial crises and stock market contagion in a multivariate time-varying asymmetric framework, J. Int. Financ. Mark. Inst. Money, 21, 1, 92-106, (2011)
[15] Kumar, M. S.; Okimoto, T., Dynamics of international integration of government securities’ markets, J. Bank. Finance, 35, 1, 142-154, (2011)
[16] Lee, R. D.; Carter, L. R., Modelling and forecasting US mortality, J. Amer. Statist. Assoc., 87, 659-671, (1992)
[17] Lee, R. D.; Miller, T., Evaluating the performance of the Lee-Carter method for forecasting mortality, Demography, 38, 537-549, (2001)
[18] Li, J. S.H.; Chan, W. S., Time-simultaneous prediction bands: a new look at the uncertainty involved in forecasting mortality, Insurance Math. Econom., 49, 81-88, (2011) · Zbl 1218.91083
[19] Li, J. S.H.; Hardy, M. R., Measuring basis risk in longevity hedges, N. Am. Actuar. J., 15, 2, 177-200, (2011) · Zbl 1228.91042
[20] Li, J. S.H.; Hardy, M.; Tan, K. S., Uncertainty in mortality forecasting: an extension to the classical Lee-Carter approach, ASTIN Bull., 39, 137-164, (2009) · Zbl 1203.91113
[21] Li, J. S.H.; Luo, A., Key q-duration: a framework for hedging longevity risk, ASTIN Bull., 42, 413-452, (2012) · Zbl 1277.91089
[22] Lin, Y.; Liu, S.; Yu, J., Pricing mortality securities with correlated mortality indexes, J. Risk Insurance, 80, 921-948, (2013)
[23] Lin, T.; Tsai, C. C.L., On the mortality/longevity risk hedging with mortality immunization, Insurance Math. Econom., 53, 580-596, (2013) · Zbl 1290.91093
[24] Lin, T.; Tsai, C. C.L., Applications of mortality durations and convexities in natural hedges, N. Am. Actuar. J., 18, 417-442, (2014) · Zbl 1414.91215
[25] Lin, T.; Tzeng, L. Y., An additive stochastic model of mortality rates: an application to longevity risk in reserve evaluation, Insurance Math. Econom., 46, 423-435, (2010) · Zbl 1231.91209
[26] Loisel, S., Serant, D., 2007. In the core of longevity risk: dependence in stochastic mortality models and cut-offs in prices of longevity swaps. Working Paper.
[27] Mitchell, D.; Brockett, P.; Mendoza-Arriaga, R.; Muthuraman, K., Modeling and forecasting mortality rates, Insurance Math. Econom., 52, 275-285, (2013) · Zbl 1284.91259
[28] Nelsen, R. B., (An Introduction to Copulas, Lectures Notes in Statistics, vol. 139, (1999), Springer Verlag New York) · Zbl 0909.62052
[29] Okimoto, T., New evidence of asymmetric dependence structures in international equity markets, J. Financ. Quant. Anal., 43, 787-815, (2008)
[30] Olivieri, A.; Pitacco, E., Stochastic mortality: the impact on target capital, ASTIN Bull., 39, 541-563, (2009) · Zbl 1179.91108
[31] Patton, A. J., Modeling asymmetric exchange rate dependence, Internat. Econom. Rev., 47, 527-556, (2006)
[32] Plat, R., On stochastic mortality modeling, Insurance Math. Econom., 45, 393-404, (2009) · Zbl 1231.91227
[33] Plat, R., One-year value-at-risk for longevity and mortality, Insurance Math. Econom., 49, 462-470, (2011)
[34] Renshaw, A. E.; Haberman, S., A cohort-based extension to the Lee-Carter model for mortality reduction factors, Insurance Math. Econom., 38, 556-570, (2006) · Zbl 1168.91418
[35] Sklar, A., Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8, 229-231, (1959) · Zbl 0100.14202
[36] Tsai, C. C.L.; Chung, S. L., Actuarial applications of the linear hazard transform in mortality immunization, Insurance Math. Econom., 53, 48-63, (2013) · Zbl 1284.91272
[37] Vogiatzoglou, M., 2010. Dynamic copula toolbox, University of Macedonoa.
[38] Wang, C. W.; Huang, H. C.; Hong, D. C., A feasible natural hedging strategy for insurance companies, Insurance Math. Econom., 52, 532-541, (2013) · Zbl 1284.91274
[39] Wang, C. W.; Huang, H. C.; Liu, I. C., Mortality modeling with non-Gaussian innovations and applications to the valuation of longevity swaps, J. Risk Insurance, 80, 775-798, (2013)
[40] Wang, J. L.; Huang, H. C.; Yang, S. S.; Tsai, J. T., An optimal product mix for hedging longevity risk in life insurance companies: the immunization theory approach, J. Risk Insurance, 77, 473-497, (2010)
[41] Wang, C. W.; Yang, S. S., Pricing survivor derivatives with cohort mortality dependence under the Lee-Carter framework, J. Risk Insurance, 80, 1027-1056, (2013)
[42] Wills, S.; Sherris, M., Securitization, structuring and pricing of longevity risk, Insurance Math. Econom., 46, 173-185, (2010) · Zbl 1231.91251
[43] Yang, S.S., Chang, Y.P., Yeh, Y.Y., 2008. A residual bootstrapped analysis of Lee-Carter model in mortality forecasting. In: 12th APRIA Annual Conference, Sydney, July 6-9.
[44] Yang, S. S.; Wang, C. W., Pricing and securitization of multi-country longevity risk with mortality dependence, Insurance Math. Econom., 52, 157-169, (2013) · Zbl 1284.91556
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